Problem: $ B = \left[\begin{array}{rr}-2 & 0 \\ -1 & 3 \\ 4 & 1\end{array}\right]$ $ F = \left[\begin{array}{rrr}-2 & 4 & 3 \\ -2 & 2 & 3 \\ 4 & 0 & 2\end{array}\right]$ Is $ B F$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ B$ , have? How many rows does the second matrix, $ F$ , have? Since $ B$ has a different number of columns (2) than $ F$ has rows (3), $ B F$ is not defined.